Many learners have difficulties moving from whole number understandings to rational number understandings. These conceptual understandings and misunderstandings become more visible as students represent that understanding in pictorial and symbolic representations of the rational number.

A critical attribute when using an area model to represent a fraction is the ability to partition the whole into same-sized parts. Students need to understand that although the area of each of the parts needs to be of equal size, these parts do not necessarily need to be of the same shape. Often during instruction students work with models that are already the same size and shape (Van de Walle, Karp & Bay-Williams, 2010). Limiting examples in this way sometimes leads to students to think that the pieces always must be the same shape. This common misconception is targeted in the Representing Fractions II assessment.

Misconception 2- Part to Whole, Equal Pieces – Does Not Consider Equivalent or Partitioned Pieces: Students with this misconception associate the number of pieces shaded with the numerator and the total number of pieces for the entire figure with the denominator. These students do pay attention to equal shape pieces but do not consider different shaped pieces with same area.

You may also want to view Student Work Samples at this time to see additional examples of responses to the assessment items.

Arrange for 10-15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the pre-assessment is designed to elicit misconceptions before instruction, you do not need to do any special review of this topic before administering the assessment.

Administering the Pre- Assessment

Inform the students about the assessment by reading the following:

"Today you will complete a short individual activity. This activity is designed to help me understand how you think about visual representations of fractions."

Distribute the assessment and read the following:

"The activity includes 4 problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about fractions, it’s important for you to include some kind of explanation in the space provided. This can be a picture or words or something else that shows how you chose your answer."

"You will have about 10 minutes to complete all the problems. When you are finished please place the paper on your desk and quietly [read, work on ____] until everyone is finished."

Monitor the students working on the assessment, making sure students understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 10 minute timeframe. It is okay if students are given more time if needed.

When a few minutes remain, say:"You have a few minutes to finish the activity. Please make sure all of your answers are as complete as possible during this time. When you are done, please place the paper face down on your desk. Thank you for working on this activity today."

When ready, collect the assessments.

After Administering the Pre- Assessment

Use the scoring guide process to analyze whether your students have Misconception 2: Part to Whole, Equal Pieces – Does Not Consider Equivalent or Partitioned Pieces

The Representing Fractions II assessment is composed of 4 items with specific attributes associated with understandings and misunderstandings related to the area model representation. We encourage you to look at the scoring guide to understand the specific attributes associated with understandings and misunderstandings related to the area model representation.

In order to determine the degree of understanding and misunderstanding, the answer to the selected response as well as the explanation text and representations must both be considered. The student work example above is one of many student work samples that provide insight into student thinking about Misconception 2 -Part to Whole, Equal Pieces – Does Not Consider Equivalent or Partitioned Pieces. We encourage you to look at more student work examples.

If administering the Representing Fractions II Pre-Assessment shows one or more of your students have this misconception, plan and implement instructional activities designed to increase student’s understanding. The post-assessment can then be used to determine if the flawed thinking supporting the misconception has been corrected.

Prior to Giving the Post-Assessment

Arrange for 10-15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the pre-assessment is designed to elicit misconceptions before instruction, you do not need to do any special review of this topic before administering the assessment.

Administering the Post- Assessment

Inform the students about the assessment by reading the following:

"Today you will complete a short individual activity. This activity is designed to help me understand how you think about visual representations of fractions."

Distribute the assessment and read the following:

"The activity includes 4 problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about fractions, it’s important for you to include some kind of explanation in the space provided. This can be a picture or words or something else that shows how you chose your answer."

"You will have about 10 minutes to complete all the problems. When you are finished please place the paper on your desk and quietly [read, work on ____] until everyone is finished."

Monitor the students working on the assessment, making sure students understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 10 minute timeframe. It is okay if students are given more time if needed.

When a few minutes remain, say:"You have a few minutes to finish the activity. Please make sure all of your answers are as complete as possible during this time. When you are done, please place the paper face down on your desk. Thank you for working on this activity today."

When ready, collect the assessments.

After Administering the Post- Assessment

Use the scoring guide process to analyze to determine if the flawed thinking supporting Misconception 2: Part to Whole, Equal Pieces – Does Not Consider Equivalent or Partitioned Pieces has been corrected.

Related Assessments: Understanding Fractions

Understanding Fractions

Representing Fractions I

Understanding Fractions

Representing Fractions II

Understanding Fractions

Locating Fractions on a Number Line

Understanding Fractions

Comparing Two Fractions

Online Administration

The EM2 Project takes full advantage of computer technologies to provide data to teachers quickly and efficiently.

About The Project

A project to develop valid and reliable, short, highly focused diagnostic assessments that are available to educators as paper assessments or through an open-source, online assessment system.

About Our Organization

Education Development Center is a global nonprofit organization with 501(c)(3) status that designs, delivers and evaluates innovative programs to address some of the world’s most urgent challenges in education, health, and economic opportunity.

About Our Funder

This website and research was supported by the IES, U.S. DOE., through Grant 305A110306 to EDC, Inc. The opinions expressed are those of the authors and do no represent views of the IES. or the U.S. DOE.